540 Mobius's Theorem on the Reversion of certain Series. 



Results of this kind do not appear to be of any great in- 

 terest, but it may be remarked that as we may obtain in this 

 manner several expressions for the same quantity f(q), we are 

 thus led to certain equalities between elliptic-functions ex- 

 pressions in which the terms are subject to Mobius's law. 



Thus, for example, from the formula 



2£K = 4S oo g^»-v 



we find 



M. = l\kK-k s K 5 -k 5 K 5 --k 7 K 1 -k ll K n -kc.}; 

 1-r? 7T ( 



2 



and from the formula 



7rY 2K V ,g. (2n-l)V»— > 



we find 



A 1 + q ir 

 From these two results it follows that 



&K 3 - 3%Kl - 5\Kl - 7%K* - &c. 

 = i7r 2 {kK-k 3 K s ^k,K 5 -k 7 K 1 -&c). 

 Similarly, since 



and 

 we find 



=^ 3 {H' 2 K 3 + 3 2 ^lK 3 -5 2 ^' 5 2 K 5 + 7 2 ^?K 7 + &c. } . 



